Accelerating Bilevel Optimization With Hierarchical Many-Threaded Parallel Differential Evolution

A Co-evolutionary Decomposition-based Chemical Reaction Algorithm for Bi-level Combinatorial Optimization Problems

Population-based optimization algorithms, such as evolutionary algorithms, have enjoyed a lot of attention in the past three decades in solving challenging search and optimization problems. In this chapter, we discuss recent population-based evolutionary algorithms for solving different types of bilevel optimization problems, as they pose numerous challenges to an optimization algorithm. Evolutionary bilevel optimization (EBO) algorithms are gaining attention due to their flexibility, implicit parallelism, and ability to customize for specific problem solving tasks. Starting with surrogate-based single-objective bilevel optimization problems, we discuss how EBO methods are designed for solving multi-objective bilevel problems. They show promise for handling various practicalities associated with bilevel problem solving. The chapter concludes with results on an agro-economic bilevel problem. The chapter also presents a number of challenging single and multi-objective bilevel optimization test problems, which should encourage further development of more efficient bilevel optimization algorithms.

A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

The Data-driven Optimization of bi-level Mixed-Integer NOnlinear problems (DOMINO) framework is presented for addressing the optimization of bi-level mixed-integer nonlinear programming problems. In this framework, bi-level optimization problems are approximated as single-level optimization problems by collecting samples of the upper-level objective and solving the lower-level problem to global optimality at those sampling points. This process is done through the integration of the DOMINO framework with a grey-box optimization solver to perform design of experiments on the upper-level objective, and to consecutively approximate and optimize bi-level mixed-integer nonlinear programming problems that are challenging to solve using exact methods. The performance of DOMINO is assessed through solving numerous bi-level benchmark problems, a land allocation problem in Food-Energy-Water Nexus, and through employing different data-driven optimization methodologies, including both local and global methods. Although this data-driven approach cannot provide a theoretical guarantee to global optimality, we present an algorithmic advancement that can guarantee feasibility to large-scale bi-level optimization problems when the lower-level problem is solved to global optimality at convergence.

Combinatorial bi-level optimization remains a challenging topic, especially when the lower-level is an NP-hard problem. In this paper, we tackle large-scale and combinatorial bi-level problems using GP hyper-heuristics, i.e., an approach that permits to train heuristics like a machine learning model. Our contribution aims at targeting the intensive and complex lower-level optimizations that occur when solving a large-scale and combinatorial bi-level problem. For this purpose, we consider hyper-heuristics through heuristic generation. Using a GP hyper-heuristic approach, we train greedy heuristics in order to make them more reliable when encountering unseen lower-level instances that could be generated during bi-level optimization. To validate our approach referred to as GA+AGH, we tackle instances from the bi-level cloud pricing optimization problem (BCPOP) that model the trading interactions between a cloud service provider and cloud service customers. Numerical results demonstrate the abilities of the trained heuristics to cope with the inherent nested structure that makes bi-level optimization problems so hard. Furthermore, it has been shown that training heuristics for lower-level optimization permits to outperform human-based heuristics and metaheuristics which constitute an excellent outcome for bi-level optimization.

In recent years, research in the field of bilevel optimization has gathered pace and it is increasingly being used to solve problems in engineering, logistics, economics, transportation etc. Rapid increase in the size and complexity of the problems emerging from these domains has prompted active interest in the design of efficient algorithms for bilevel optimization. While Memetic Algorithms (MAs) have been quite successful in solving single level optimization problems, there have been very few studies exploring their application in bilevel problems. MAs essentially attempt to combine advantages of global and local search strategies to locate optimum solutions with low computational cost (function evaluations). In this paper, we present a new nested approach for solving bilevel optimization problems. The presented approach uses memetic algorithm at the upper level, while a global or a local search method is used in the lower level during various phases of the search. The performance of the proposed approach is compared with two established approaches, NBLEA and BLEAQ, using SMD benchmark problem set. The numerical experiments demonstrate the benefits of the proposed approach both in terms of accuracy and computational cost, establishing its potential for solving bilevel optimization problems.

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to varepsilon -feasibility regarding its nonlinear constraints for an arbitrarily small but positive varepsilon , the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater’s constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of varepsilon -feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.

K-means cluster interactive algorithm-based evolutionary approach for solving bilevel multi-objective programming problems

Presolving linear bilevel optimization problems

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

Bilevel optimization involves two levels of optimization, where one optimization problem is nested within the other. The structure of the problem often requires solving a large number of inner optimization problems that make these kinds of optimization problems expensive to solve. The reaction set mapping and the lower level optimal value function mapping are often used to reduce bilevel optimization problems to a single level; however, the mappings are not known a priori, and the need is to be estimated. Though there exist a few studies that rely on the estimation of these mappings, they are often applied to problems where one of these mappings has a known form, that is, piecewise linear, convex, etc. In this article, we utilize both these mappings together to solve general bilevel optimization problems without any assumptions on the structure of these mappings. Kriging approximations are created during the generations of an evolutionary algorithm, where the population members serve as the samples for creating the approximations. One of the important features of the proposed algorithm is the creation of an auxiliary optimization problem using the Kriging-based metamodel of the lower level optimal value function that solves an approximate relaxation of the bilevel optimization problem. The auxiliary problem when used for local search is able to accelerate the evolutionary algorithm toward the bilevel optimal solution. We perform experiments on two sets of test problems and a problem from the domain of control theory. Our experiments suggest that the approach is quite promising and can lead to substantial savings when solving bilevel optimization problems. The approach is able to outperform state-of-the-art methods that are available for solving bilevel problems, in particular, the savings in function evaluations for the lower level problem are substantial with the proposed approach.

Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve—both in theory and practice. Fortunately, there have been significant algorithmic advances in the field of bilevel optimization so that we can solve much larger and also more complicated problems today compared to what was possible to solve two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. This survey gives a detailed overview of one of these more challenging classes of bilevel problems: bilevel optimization under uncertainty. We review the classic ways of addressing uncertainties in bilevel optimization using stochastic or robust techniques. Moreover, we highlight that the sources of uncertainty in bilevel optimization are much richer than for usual, i.e., single-level, problems since not only the problem’s data can be uncertain but also the (observation of the) decisions of the two players can be subject to uncertainty. We thus also review the field of bilevel optimization under limited observability, the area of problems considering only near-optimal decisions, and discuss intermediate solution concepts between the optimistic and pessimistic cases. Finally, we also review the rich literature on applications studied using uncertain bilevel problems such as in energy, for interdiction games and security applications, in management sciences, and networks.

The bilevel programming problem (BLP) is an optimization problem with another optimization problem in its constraints. This framework finds utility in modeling decentralized scenarios, which arise in real-world applications such as traffic management, transportation, and economic policy. Differential Evolution (DE) techniques have emerged in literature for addressing such complex problems. However, handling linear equality constraints poses a significant challenge for DE and other metaheuristics. To address this issue, we previously introduced DELEqC, enhancing DE with a mechanism to manipulate the linear equality constraints. A specialized variant, BL-DELEqC, was further proposed specifically for tackling general BLPs. Another variant, DELEqC-III, transforms the original constrained optimization problem into a lower-dimensional unconstrained one, offering applicability to BLPs with linear equality constraints. Thus, we explore in this study the efficacy of DELEqC-III in handling BLPs with linear equality constraints. The proposed BL-DELEqC-III is compared to BL-DELEqC on a selection of benchmark BLPs, demonstrating superior results.

Bilevel optimization, as the name reflects, deals with optimization at two interconnected hierarchical levels. The aim is to identify the optimum of an upper-level leader problem, subject to the optimality of a lower-level follower problem. Several problems from the domain of engineering, logistics, economics, and transportation have an inherent nested structure which requires them to be modeled as bilevel optimization problems. Increasing size and complexity of such problems has prompted active theoretical and practical interest in the design of efficient algorithms for bilevel optimization. Given the nested nature of bilevel problems, the computational effort(number of function evaluations) required to solve them is often quite high. In this article, we explore the use of a Memetic Algorithm (MA) to solve bilevel optimization problems. While MAs have been quite successful in solving single-level optimization problems, there have been relatively few studies exploring their potential for solving bilevel optimization problems. MAs essentially attempt to combine advantages of global and local search strategies to identify optimum solutions with low computational cost(function evaluations). The approach introduced in this article is a nested Bilevel Memetic Algorithm(BLMA). At both upper and lower levels, either a global or a local search method is used during different phases of the search. The performance of BLMA is presented on twenty-five standard test problems and two real-life applications. The results are compared with other established algorithms to demonstrate the efficacy of the proposed approach.

Both pessimistic and optimistic bilevel optimization problems may be not stable under perturbation when the lower-level problem has not a unique solution, meaning that the limit of sequences of solutions (resp. equilibria) to perturbed bilevel problems is not necessarily a solution (resp. an equilibrium) to the original problem. In this paper, we investigate the notion of lower Stackelberg equilibrium, an equilibrium concept arising as a limit point of pessimistic equilibria and of optimistic equilibria of perturbed bilevel problems. First, connections with pessimistic equilibria and optimistic equilibria are obtained in a general setting, together with existence and closure results. Secondly, the problem of finding a lower Stackelberg equilibrium is shown to be stable under general perturbation, differently from what happens for pessimistic and optimistic bilevel problems. Then, moving to the game theory viewpoint, the set of lower Stackelberg equilibria is proved to coincide with the set of subgame perfect Nash equilibrium outcomes of the associated Stackelberg game. These results allow to achieve a comprehensive look on various equilibrium concepts in bilevel optimization and in Stackelberg games as well as to add a new interpretation in terms of game theory to the previous limit results on pessimistic equilibria and optimistic equilibria under perturbation.